Question

If the generation time of a bacterium is 40 minutes and a culture containing \[{{10}^{7}}\] cells / ml is grown for 4 hours, then calculate its population after that period:
(a) \[64\times {{10}^{7}}\]
(b) \[32\times {{10}^{7}}\]
(c) \[6\times {{10}^{7}}\]
(d) \[40\times {{10}^{7}}\]

Answer 1

Hint: Convert hours into minutes by multiplying the given hour with 60. Use the information that a bacteria splits into two parts after 40 minutes and calculate the total number of bacteria formed after 4 hours. Calculate the value for 1 bacteria and multiply with \[{{10}^{7}}\] to get the correct option.


Complete step by step answer:
Here, we have to find the total population of bacteria after 4 hours if one bacteria splits into two parts after each 40 minutes. We have a total of \[{{10}^{7}}\] bacteria.
Let us calculate the total population formed from 1 bacteria.
We have 4 hours given.
We know that, 1 hour = 60 minutes.
\[\Rightarrow \] 4 hours = \[4\times 60\] = 240 minutes.
Now, the population of bacteria formed after 40 minutes = 2.
After 40 \[\times \] 2 = 80 minutes, population = 2 \[\times \] 2 = \[{{2}^{2}}\].
After 40 \[\times \] 3 = 120 minutes, population = \[{{2}^{2}}\times 2={{2}^{3}}\].
After 40 \[\times \] 4 = 160 minutes, population = \[{{2}^{3}}\times 2={{2}^{4}}\].
After 40 \[\times \] 5 = 200 minutes, population = \[{{2}^{4}}\times 2={{2}^{5}}\].
After 40 \[\times \] 6 = 240 minutes, the population becomes = \[{{2}^{5}}\times 2={{2}^{6}}\].
So, after 240 minutes or 4 hours 1 bacteria will form \[{{2}^{6}}\] = 64 bacteria. Since, we have \[{{10}^{7}}\] bacteria in total, so each one will split into 64 bacteria. Therefore, we have,
Total bacteria formed from the culture = \[64\times {{10}^{7}}\].

So, the correct answer is “Option A”.

Note: One may note, we have converted 4 hours into 240 minutes. One can also apply the reverse process by converting 40 minutes into required hours by dividing it with 60 and then calculate the population after 4 hours. We have first calculated the population for 1 bacteria because if we will directly consider \[{{10}^{7}}\] bacteria then we may get confused.